Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9

Validated structural analysis of Gothic vaulted systems

E. Erdogmus & T.E. Boothby The Pennsylvania State Un iversity, University Park, USA

ABSTRACT: A validated analytical procedure for vaulted unreinforced masonry structures is proposed in this study. The structural analyses necessary in this research are carried out by the finite element analysis method (FEM). The ribs of a gothic rib vault are modeled using linearly elastic solid elements, while the vault webs are modeled using linearly elas ti c shell elements. The analytical models are then validated using in situ nondestructive vibration experiments, by the comparison of analytical and experimental mode shapes and natural frequencies ofthe vaulting system. Appropriate boundary conditions are determined by the verification ofmode shapes, and material properties are verified by comparison of natural frequencies. National Cathedral in Washington, DC, U.s.A. is studied as a controlled case as it has known material properties and construction history. These analysis and validation methods studied on NC will be then applied to European medieval structures in the continuation of this research.

INTRODUCTION

Gothic architecture initiated in France in the 12th cen­tury and spread through Europe for four centuries. Within the last few hundred years, other structures around the world such as the National Cathedral (NC) in Washington, DC were also constructed in Gothic style. The main component in these unrein­forced masonry structures is the complex assembly of vaulted roof systems, for which, there are not well­established and dependable analysis techniques. The available methods are rigid-plastic analyses, elastic analyses, two-dimensional graphical analyses and a current trend of computer analyses mostly based on the finite element method. However, none of these meth­ods has been validated by experiments conducted on the real structures (Boothby, 200 I) .

A validated analytical procedure for vaulted sys­tems is evident is proposed in this study. The struc­tural analyses necessary in this research are carried out using three-dimensiona l finite element analy­sis method (FEM). The analytical models are then validated using in situ nondestructive experiments.

For the case of a Gothic vaulted roof struc­ture, besides the complexity of the structural behav­ior, material properties and support conditions are unknown. Static load testing or sample extraction, which would reveal information about these param­eters are not possible on these historically impor­tant structures. Therefore, nondestructive testing is called upon for the validation of FEM. In this study, in situ vibration experiments and experimental modal

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analyses are used for the updating and validation of the mode!.

The proposed methods are first studied on a struc­ture with better-known properties and history, the National Cathedral (NC). Therefore, the ribbed vaults in the choir of NC are modeled, the analytical mod­eIs are experimentally validated, and the results ofthis study are presented in this papeI. The findings, results and improvement suggestions from this study will be applied to similar studies on medieval European structures in the future.

1.1 Washington National Cathedral

The construction of the NC, the sixth largest cathe­dral in the world, continued for more than 83 years (September 29, 1907 to September 29, 1990). Frederick Bodley, England's leading Anglican Church architect, was the first head architect with Henry Vaughan as the supervising architect. After Bodley and Vaughan had passed away following World War I, American architect Philip Hubert Frohman became the principal architect. The cathedral was built in the Decorated Gothic style, a style in English architecture that lasted from 1270 to 1380. The cathedral was engi­neered and constructed like the medieval churches, using only quarried stone with no steel reinforcement in any part ofthe building and no mass-produced ele­ments. Even though the exact dates are unknown, the choir vaults, which will be analyzed in this srudy, were finished around the 1930's, when the transepts were opened for public use. The even-Ievel-crown,

Figure I . Tierceron vaults in the choir of National Cathedral.

Tierceron rib vaults, which cover the rectangularplan bays of the church, are constructed using Indiana limestone (Fig. I) (www.cathedral.org; Ed. Fallen, 1995).

2 FINITE ELEMENT MODEL CREATION

The finite element model creation starts with the geo­metric modeling. The models ofNC are created using the scaled drawings from the NC archives (National Cathedral Archives) and on-site observation.

After the geometric model is created, numerous unknown parameters are added to the model for a structural analysis, such as the material properties and the boundary conditions. These parameters affect the structural behavior ofthe model greatly and it is impos­sible to have confidence in these parameters without verification from some experimental data. For that reason, some experiments on the vaults ofNC are con­ducted, which reveal information about the structural behavior. Once these parameters are updated in the analytical models based on experimental findings, the analyses to be performed on these models will be more reliable. The final boundary conditions and the mate­rial properties for this model, as updated by the experi ­mentai results are presented in the subsequent sections.

2.1 Finite element model material properties

The models are created with the assumptions of lin­ear behavior and homogenous material. The initial assumption for linear behavior is proved to be appro­priate afier the review of dynamic experiment results, because the impacts from different magnitudes of impact (heel drops vs. hammer impact) generate simi­lar behavior on the structure, as will be presented later

Figure 2. The National Cathedral three-vau lt fini te ele­ment model (software: ANSYS) and the experimental setup schematic shown on the fin ite element model.

Table I. Masonry assembly stiffness values.

Joint Mortar thickness Masonry Eclf (E in GPa) (mm) (in) (E in GPa) (GPa)

Max 1.4 6.4 (1/4) 31 21 1.4 9.5 (3/8) 31 \ 8

Average 0.85 6.4 ( \/4) 3 1 \8 0.85 9.5 (3/8) 3 \ \5

Min 0.3\ 6.4 ( \/4) 3 \ lO 0.31 9.5 (3/8) 3\ 8

in this paper. The homogeneous material assumption, which considers effective material properties fo r the masonry unit and mortar assembly rather than sepa­rate entities for units and masonry joints, is adopted for simplification ofthe fin ite element analysis procedure.

The material properties are gathered from various sources such as literature (www.langstone.com). lab­oratory experiments on an Indiana lime stone sample, and calculations to account for the mortarunit assem­bly assumption. Depending on the stiffness ofthe mor­tar and the thickness of the mortar joint, the effective stiffness of the assembly varies. Table I summarizes the possible Modulus of Elasticity (E-value) ranges fo r masonry assembly with type O mortar and Indiana lime stone. The mortar E-values are from McNary & Abrams (1985) and E value for the Indiana lime stone is from laboratory testing. The mortar joint thicknesses vary between 9.5 mm and 6.4mm (3/8"-1/4").

According to the information given in Table I, the modulus of elasticity for the masonry-mortar assembly for the finite element model ranges between 8 GPa to 21 GPa, depending on the condition and thickness of the mortar joints. When the model is updated by the experimental results, the optimum value for the masonry assembly stiffness is found to be 12 GPa. This is partially due to the fact that the masonry in the vaults

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Table 2. Final material properties for NC finite element model.

Modulus of elasticity Density Poisson 's

Material (GPa) (kg/m3) ratio

Masonry assembly 12 2100 0.2 Concrete surcharge 24 2100 0.15

Figure 3. Final boundary conditions as updated by the experimental findings.

has lost some ofits initial stiffness due to loss ofmortar and cracks at some joints, which results in an E-value at the lower end of the range.

The other material properties required are the den­sity and the Poisson's ratio of the masonry assembly, and the material properties for the surcharge, which is concrete in the case of NC. Table 2 presents the experimentally validated final values for these parameters.

2.2 Finite element model boundwy conditions

The boundary conditions are applied based on obser­vations and are then updated by experimental results to match the experimentally observed mode shape. The final boundary conditions, as illustrated in Fig­ure 3, are as follows: The sides of the arches are restrained in the Y direction to account for the adja­cent vaults. The top portion of the side arches is restrained in the vertical direction to account for the vertical-movement-limiting effects ofthe adjacent arches. Parts of the exterior wall are restrained in the X direction to account for the wall buttresses, flying

buttresses and connection of the roof trusses to the exterior walI.

3 EXPERlMENTAL PROCEOURES ANO RESULTS

In this research, the experimental modal analysis techniques are used to validate the finite element models of the complex vaulted systems. This is an application of dynamic experiments that the design of experiments constitutes an important part of the procedure. Therefore, experience from similar experi­ments performed on other types of structures is called upon. Hanagan et aI. (2003) investigate the variables that affect experimental moda I analysis of floor sys­tems, some of which are discussed in the following sections and are used in this study to determine the experimental procedures.

3.1 Experimental equipmenf and se fup

In this study, three different excitation methods are used: heel drops, shaker excitation and hammer impacts. The heel drop is a man-made impact created by raising up on one's toes and dropping the heels onto the floor. The heel drop impact force is measured using a force plate constructed by attaching four shear beam load cells to the corners of a thick aluminum plate (Hanagan , et aI. , 2003). The force plate used for heel drops in this study, has a calibration coefficient of 3916 N/volt. The instrumented hammer is another convenient tool to excite a floor system because of its portability. The instrumented hammer used is PCB Piezetronics ICp® lmpulse Hammer Model 086C20. The hammer has options for creating the impact, including a selection of tips with varying hardness. Since the hammer is instrumented, the force plate is not required and the hammer can be connected directly to the data acquisition system. However, in this set of experiments another custom designed force plate, with a calibration coefficient of 80 I N/volt is used to measure the force created by the hammer.

The force plate, which is used for the heel drops, is also used for the shaker. Oue to the heavy weight and low coherence caused by some technical problems with the available shaker at hand, the shaker is not used in these experiments extensively. However, as better data is available from shaker excitation for one ofthe setups (Row4), the data from that experiment is used for determining the mo de shape and is presented in this paper as evidence for the consistency ofthe results regardless of the excitation type and thus the linearity of the structural behavior.

Seven PCB Piezetronics model 393A03 seismic accelerometers, with a sensitivity of I V/g are used as vibration transducers and were placed in vari­ous arrays. The experiments on the NC vaults cover

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three bays of the choir with the driving point on the crown of the third bay counted from the crossing. An experimental setup plan and the picture from the exper­imentaI site are shown in Figure 4. A schematic rep­resentation of the experimental setup is also provided on the finite element model in Figure 2.

Other variables that affect experimental modal anal­ysis offloor systems as investigated by Hanagan et a!. (2003), and are pertinent to this study are briefly discussed below.

A SIGLAB dynamic signal analyzer is used fo r col­lecting and processing the data. This dynamic signal analyzer includes anti aliasing filters to eliminate the high frequency signals and allows for real-time post-processing and adjustments.

2 For efficient data processing, considerations such as the selection of the record length, bandwidth, averaging and windowing are used to achieve a good frequency resolution, to adequately capture the data range of interest, to distinguish the data from the

~ ..

Figure 4. The experimental setup shown on the partial floor plan and the photo of the experiment site above the vaults.

ambient noise and to avoid leakage, respectively. Ali ofthese are set-up using the SIGLAB software graphical interface.

3.2 Experimental data analysis and results

The parameters that can be extracted through vibration experiments and would be useful for comparison with analytical models are the mo de shape, the frequency for the observed mode and an approximation for the system stiffness. Consequently, two types of exper­imentaI outputs are of concern: frequency response functions (FRF) and Coherence (CO H). FRF is the complex function of the ratio between a harmonic displacement response and the harmonic force in the frequency domain (Raebel, 2000). Using a force plate or the load cell within the hammer, the force input is measured and thus the FRF is plotted in terms of the normalized acceleration amplitude. Since it is a com­plex function, an FRF can be expressed by the real and imaginary parts, or in polar coordinates as the phase angle and magnitude. Ali of these possible presenta­tions of an FRF revea! different information about the moda! behavior of the structure.

The magnitude and phase plots ofthe FRF reveal the resonant peaks ofthe response and the corresponding 90° phase shift at the natural frequency ofthe observed

NC_ACoW_HeeIOrops Accelerance FRF & Coherence

:.:: 1--:::7"''\ir-.......... -,..,-''''IIilIIiI,.".:tiij~l o.,

3.5OE..()6

3,OOE.os

~ 2.50E..()6

i 2.00E-OO

1.5OE-06

5.00e-07

o .• 0.1

~ 0.6 ~

0,5 ~ 0 .4 U

0.3

0.2

o., O.OOE-OO lL __ ~===--_________ --' o

o 10 15 20 2S 30 35 40 45 50

Frequenc:y (lU)

Figure 5. FRF magnitude and coherence plot for heel drops for lhe NC experiment setup col4.

90

60

Ui' 30

" O i!! Dl -30 " E.

-60 " i(l -90 .r:

'" -120

-150

-180

NC_ACoI4_HeeIDrops Phase Piai

20 25 30 I

Frequency (Hz)

Figure 6. FRF phase plot for the NC experiment setup col4.

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mode ofthe system. Figures 5 and 6 present these plots for the heel drop tests in the NC, for the sensor located immediately adjacent to the driving point for the setup shown in Figure 4. As can be seen from both plots, there is an apparent mode with a natural frequency of36Hz.

Since the force input is normalized and plotted for every frequency, FRF allows for averaging and com­paring data from severa I excitations even though the applied force intensity varies. The coherence function measures how well the output is linearly related to the input for every frequency, and is rated on a O to I scale. Coherence measurements approaching unity indicate a linear trend between output and input, and in most cases represent high confidence in the measured data (Hanagan et aI., 2003; Raebel, 2000). The coherence for the heel drop measurements for the sensor next to the driving point is presented in Figure 5 and it suggests good quality data except the 0- 5 Hz low fre­quency range, where the ambient noise interferes with the generated vibrations.

Figure 7 shows the FRF from a hammer impact at the same driving point. As can be seen, the low coherence for hammer impact covers a larger range of lower frequencies (0- 25 Hz). This is mainly because the hammer impact is a low amplitude excitation, that can generate lower frequency vibrat ions and therefore the data is more susceptible to the ambient noise. On the other hand, the FRF plots for these two differ­ent excitation leveis are very similar, which supports the assumption of linear behavior for this structural system.

The detlected shape ofthe structure from the ex per­imentally observed mode shape is gathered from the imaginary plots, as this representation ofthe FRF also includes the sense of the acceleration for each sen­sor (Figs 8 & 9). By noting the positive and negative motion of each sensor, the deformed shape for the test­ing scheme illustrated in Figures I and 4 is plotted in Figure 10.

By comparing this experimentally observed deformed shape with the analytically established mode

4 SE-06 ,--------'==5j,;,;;;. ____ --..... 4.0E·06

35E-06

3.QE.()6

~ 2.5E-06

~ 2.0E-06

'" 1.5E-06

O. 08

07

0.6 ~

0.5 ~

0.4 ;3 o.,

I .OE-06 02

SOE..Q7 01

0.0'>00 L---_..:.:::::===~ _________ --1 10 15 20 25 30 35 40 4S 50

Frequeney (Hz)

Figure 7. FRF magnitude and coherence plot for hammer impacI for the NC experiment setup col4.

shapes, it is possible to validate the boundary condition assumptions, because the mode shape is determined mainly by boundary conditions. Other parameters such as the material properties (Masonry and surcharge stiffness, density and Poisson 's ratio) have no effect on the mode shape, although they affect the natural frequency of this mode shape.

An additional experimental result of interest is the FRF plotted in displacement units, because the displacement at zero frequency gives the inverse of the system stiffness, i.e. the system tlexibility. Since the sensors measure acceleration, the data collected must be converted to displacement units

NC_ ACo14_Heel Orop FRF _Imaginary Plol

1.E-06.,----,----,--------,-------.--.

-1 .E·06 - Al

~ -2E-06 -~; ,. - -1- - - - - - ,. - ::- - - -1- -

-3.E-06 ~~~ f--:------+-~----:---4.E-06 - A6 ..l. - -1- - - - - _..l. _..J_ - - _1 __

A7 1 1 1 -S.E-06 "-'===---------------'

Frequency (Hz)

Figure 8. Imaginary plot for heel drop excitation for ali sensors in the NC experiment setup col4.

NC_ARow4_ Shaker Imaginary Plol

1.0E-04 r--------=----'--'-------, S.OE-OS

~ O.OE+OO r""':'-~"""":~~~~IIIIIiB:"'::';r2~~~~ -; -SOE-OS / S P _ l? _ lfJ _ _ 2~ __ o - Al i 1 1 ~ -1.0E-04 - A2 ,. -,- -,- - - --

~ _ 1 .SE-04 A3 ~A4 1 1 1

~ -2.0E-04 - AS I' - 'I - -1- - - - - -

-2.SE-04 - A7 1- - -j - -1- - - - ---3.0E-04 +>==C=h=S'-'-_-'-----' ________ L--1

Frequency (Hz)

Figure 9. Imaginary plot for shaker excitation for ali sensors in the NC experiment setup row4.

Figure 10. Defonned shape of the top beams detected by the grid of accelerometers aI 36 Hz.

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NC_ACoI4_Heel Orop. Compliance FRF

4.0E-Oa -rr--~---":"""------~-----, Z 3.5E-Oa - - - -L - - - - - - - - - - - - - - -I - -Ê 3.0E-Oa - - - .1 - - - - - - - - - - - - - - _I - -

E 2.5E-oa - + ---------~ 2.OE-oa - - T - - -

1l 1.5E-oa -ã. 1.OE-oa 6 5.0E-09 -r..il:'o-1~~""",,,,,""'rdI"

O.OE+OO +--_---l-_-_-_~-__ -_+_~ o 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Figure 11_ Displacement FRF for the NC experiment setup col4.

using the relationship between the acceleration and displacement as shown by Equation I.

X=-o/X X or X=-­

-0/

Where, X is the acceleration X is the displacement w is the circular natural frequency

(I)

The data collected by the accelerometer next to the driving point during heel drops is eonverted to dis­placement units by Equation I, and plotted as shown in Figure 11.

If the aceeleration at zero frequeney were aetually zero, then the equation would result in a singularity and would be solved for by an indefinite integral solu­tion. Thus, the eonversion between the acceleration and displaeement units in theory is not trivial but it is possible. However, in praetice, this eonversion gener­ates some problems. The ambient noise results in very small vibration amplitudes at the lower frequeneies, whieh is also apparent from the low eoherence between 0-5 Hz, as shown in Figure 5. Therefore, the displace­ment plot is not reliable at these lower frequencies. But ifa displacement FRF for a simple system is inspected, it is observed that there is a plateau starting at about the root of the resonant peak of a natural frequeney and ending at O Hz. Therefore, the value at 25 Hz can be extrapolated in the plot in Figure lOto O Hz and hence result in an approximate value for the system stitfness of 5E-9. Although approximate, this system stitfness value can then be eompared to analytically established values to see if they are in the same order of magnitude.

4 MODEL VALIDATION

The finite element model is analyzed by linear elas­tie, undamped moda I analysis for a comparison with the experimental results. The modal analysis equation

1I!~",1

.\I&"S P',IW")~. 52' Vl ( ... >'tl ~lfi .. Q (+I)' ·.(ttl~U

~"~,(l<l"'" !)!KlI ".!lIlZl:;t't

..~H"~ _,010.·"'5

ti/lti. o :a:' C/ltta<:tr.' K1,Ilt.i.J'l. va"l't ~"d.l

"'fI" 10 ::OIJ ~ u.~.'o:

Figure 12. The deformed shape and vertical displacements plot on the full three-vault model.

used by the finite element analysis software ANSYS is given by Equation 2 (ANSYS, 2003).

Where, [KJ is the stiffness matrix {cp;} is the mode shape vector (eigenvalue) ofmodei W; is the natural circular frequency ofmode i w? is the eigenvalue [MJis the mass matrix

(2)

When such an analysis is performed on the three­vault model ofNe, the 5th mode ofvibration, which is the lowest mode with a substantial vertical movement at the sensor points, presents a similar deformed shape as the experimentally observed one, at 35.5 Hz. These results are for the previously described final material properties and boundary conditions, and present a very good agreement with the experimental results. Hence the model is validated with the in situ experiments. The results for this mode shape are shown in Figure 12 on the full model and Figure J3 shows the deformed shape for the top ribs only.

IfFigure 13 is compared to Figure 10, it is seen that the analytically determined mo de shape for the top ribs are in very good agreement with the experimentally observed deformed shape.

In order to determine the system stitfness analyti­cally, a static unit load is applied to the point, which is the driving point during the experiments and the dis­placement at this point is sought. Figure 14 presents the graphical results for this static analysis.

The system flexibility (inverse of system stiffness) determined by this analytical model is 4E-09 mlN while the experimentally gathered approximate value is 5E-09 mIN. The two are in the same order of mag­nitude. The error is due to the low coherence of

426

n.,.l lIv..j ",.-n.U6 \I: tA""J 1'1 ...... 0 ()IU· .QoOlU' IHII • • . OO,.~.,

"I ".Olll'~'

.o.PI\.102COt I j,.),,,

Figure 13 . The deformed shape and vertical di splacements illustrated only on the top beams of the three-vault model.

""1 ... . 1'. ·0' .ui •. t u a 07 Im •. • H ! "ot

.0.'1\.1 0200 , 1,: .,:.8

Figure 14. The static analysis with unit load for system stiffness.

the experimental data at the lower frequencies and the errors emerging from the manual post-processing conducted to gather the displacement data from the acceleration data.

5 SUMMARY AND DISCUSSIONS

In this study, the experimentally observed mode shape of the tested structure and the natural frequency of this mode are compared to the analytical model results for the validation of analytical models. As the applied boundary conditions result in a mode shape that is in very good agreement with the experimen­tally observed mode, the next step is to determine the most accurate material properties within the appropri­ate range, which is gathered through a combination of lab experiments and literature. Effective stiffness values are calculated to account forthe combined prop­erties of the Indiana lime stone and type O mortar.

With these material properties, the analytical natural frequency is in very c10se agreement with the exper­imentally recorded natural frequency. The analytical system stiffness is also compared to an approximate experimental value, which are in the same order of magnitude. Although there is 25% error in this com­parison, this error is mostly due to the approximately calculated value ofthe displacement at zero frequency. As a result of ali three types of comparisons and updat­ing, the analytical model is successfully validated by the nondestructive vibration experiments.

Besides proposing a validation method for the complex three-dimensional finite element models of the stone vaulted systems, this study reveals severa! useful facts about the dynamic testing of such struc­tures. For instance, it is observed that the heel drops give more coherent results than the impulse hammer impacts. However, as heel drops are not possible on most vaulted systems, an instrumented hammer, which presents the advantages of convenience and portabil­ity, is recommended, since it gives comparable results for the mode shape and natural frequency.

Another finding through various experimental tri ­ais in this study is that, the structure behaves linearly under dynamic loading and thus linear analysis on the structure is appropriate.

The work presented in this study is an excerpt from an ongoing research project, but it reveals some very useful conc1usions for the future work of the study, which explores the extrapolation ofthe proposed methods to European medieval structures. For these cases where the material properties are also unknown, the vibration experiments on the structure itself and the knowledge from similar structures with calibrated models, such as established here for NC, can be used for model validation.

ACKNOWLEDGEMENTS

The authors would like to thank to Professor Linda Hanagan, the National Cathedral administration, Joe Alonso and Professor Robert Mark for their support and patience. The assistance of Guoliang Jiang and Carlos Gomez are also greatly appreciated.

REFERENCES

ANSYS Incorporation, (2003) . ANSYS Elements Reference. Boothby, T.E. (2001). "Analysis of Masonry Arches and

Vaults," Prog. Struc/. Engng Mater. Vol 3, 246-256. Hanagan, L.M., Raebel , C.H. , and Trethewey, M.W. (2003).

"Dynamic Measurements of In-Place Steel Floors to Assess Vibration Performance." Journal of Pelformance ofConstructed Facilities, ASCE, 17(3), 126- 135.

http://www.cathedral.org http://www.langstone.com/i ndiana.htm

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McNary, W.S., and Abrams, D.P. (1985), "Mechanics of Masonry in Compression," Journal o( Structural Engi­neering, ASCE, Vol. 111 , No. 4, 857- 870.

Raebel, e.H. (2000). "Development ofan experimental pro­tocol for floor vibration assessment," Masters Thesis, Pennsylvania University.

Washington National Cathedral Guidebook, (1995) Ed. Anne-Catherine Fallen, Research & Design, Lld., Washington National Cathedral, Washington.

Washington National Cathedral Archive, Washington National Cathedral, Washington, De.

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